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140 3 Image processing

(c)

(a)

(b) (d)

Figure 3.40 Steerable shiftable multiscale transforms (Simoncelli, Freeman, Adelson et al. 1992) c 1992 IEEE:
(a) radial multi-scale frequency domain decomposition; (b) original image; (c) a set of four steerable filters; (d)
the radial multi-scale wavelet decomposition.

Their representation is not only overcomplete (which eliminates the aliasing problem) but is
also orientationally selective and has identical analysis and synthesis basis functions, i.e., it is
self-inverting, just like “regular” wavelets. As a result, this makes steerable pyramids a much
more useful basis for the structural analysis and matching tasks commonly used in computer
vision.
Figure 3.40a shows how such a decomposition looks in frequency space. Instead of re-
cursively dividing the frequency domain into 2 × 2 squares, which results in checkerboard
high frequencies, radial arcs are used instead. Figure 3.40b illustrates the resulting pyramid
sub-bands. Even through the representation is overcomplete, i.e., there are more wavelet co-
efficients than input pixels, the additional frequency and orientation selectivity makes this
representation preferable for tasks such as texture analysis and synthesis (Portilla and Simon-
celli 2000) and image denoising (Portilla, Strela, Wainwright et al. 2003; Lyu and Simoncelli
2009).

3.5.5 Application: Image blending
One of the most engaging and fun applications of the Laplacian pyramid presented in Sec-
tion 3.5.3 is the creation of blended composite images, as shown in Figure 3.41 (Burt and
Adelson 1983b). While splicing the apple and orange images together along the midline
produces a noticeable cut, splining them together (as Burt and Adelson (1983b) called their
procedure) creates a beautiful illusion of a truly hybrid fruit. The key to their approach is
that the low-frequency color variations between the red apple and the orange are smoothly